Effects of the Initial Mass Function and Mass Segregation in the Evolution of Embedded Star Clusters .

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Facultad de Ciencias Físicas y Matemáticas - Programa de Magíster en Ciencias con mención en Física

Eects of the Initial Mass Function and

Mass Segregation in the Evolution of

Embedded Star Clusters

(Efectos de la Función de Masa Inicial y

Segregación de Masa en la Evolución de

Cúmulos Inmersos en Gas)

Tesis para optar al grado de Magíster en Ciencias con mención en Física

Por

Raúl Esteban Domínguez Figueroa

Concepción, Chile Marzo 2017

Profesor Guía: Dr. Michael Fellhauer

Departamento de Astronomía, Facultad de Ciencias Físicas y Matemáticas Universidad de Concepción

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Quiero agradecer en primer lugar a mis padres Raúl Domínguez y Yolanda Figueroa por apoyarme sobre todo en difíciles momentos de este largo camino . Les agradezco a mis compañeros que siempre han tenido una palabra de apoyo que me ayudaron a seguir adelante.

Este trabajo no hubiera podido ser posible sin la ayuda y apoyo de mi supervisor Michael Fellhauer, el cual siempre conó en mi y estuvo impulsándome a mejorar.

No puedo dejar de mencionar a los post-doc Rory Smith, Graeme Candlish and Jörg Dabringhausen que siempre estuvieron dispuestos a ayudar. También agradezco a Juan Pablo Farías el cual se mantuvo siempre atento a cualquier duda que tuviera. No puedo olvidar a Dominik Schleicher mas que como co-supervisor, lo consideré mas como un amigo. Gracias también a Amelia Stutz que aunque fue poco el tiempo que compartimos siempre estaba ahí ofreciendo su ayuda.

Gracias además al aporte nanciero recibido por parte Fondecyt, Basal y SOCHIAS, que hicieron posible cubrir gastos de arancel y viajes.

No todo fue trabajo, así que agradezco a la party-woman Annette Gerlach que siempre tenía una sonrisa para regalar o una cerveza algunas veces!.

No puedo terminar sin antes decir gracias a mi polola Carla Fuentes por el continuo soporte y alegría luego de esas largas noches de trabajo sin dormir.

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Contents

Agradecimientos ii

List of Figures vi

List of Tables xiii

Resumen xiv

Abstract xv

1 General Introduction 2

1.1 Types of Star Cluster . . . 2

1.1.1 Associations . . . 3

1.1.2 Embedded Star Clusters . . . 4

1.1.3 Open Star Clusters . . . 5

1.1.4 Globular Clusters . . . 6

1.2 Hertzsprung-Russell diagram . . . 8

1.3 Overview of Star Formation Process . . . 9

1.4 Virial Ratio . . . 10

1.5 The Initial Mass Function . . . 10

1.6 Internal Processes of Young Star Clusters . . . 12

1.6.1 Timescales . . . 12

1.6.2 Mass Segregation . . . 13

1.6.2.1 Radial Mass FunctionsMMF . . . 14

1.6.2.2 Minimum Spanning Tree ΛMST . . . 14

1.6.2.3 The Local Density RatioP LDR . . . 15

1.6.2.4 Comparison of the Methods . . . 15

1.6.3 Overview of Star evolution . . . 24

1.6.3.1 Evolution of High Mass Stars . . . 24

1.6.3.2 Evolution of Low-Mass Stars . . . 26

1.6.3.3 Final-Stage of Stars . . . 27

1.7 Infant mortality . . . 28

2 Codes to use 29 2.1 IMF Generator . . . 29

2.2 Fractal Generator . . . 31 iii

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2.3 Nbody6 . . . 33

2.3.1 Gas Cloud . . . 34

2.3.1.1 Uniform Potential . . . 34

2.3.1.2 Plummer Potential . . . 34

2.4 Bound Mass Measurement . . . 36

3 Mass Segregation Evolution 38 3.1 Aim . . . 38

3.2 Previous Works . . . 38

3.3 Method . . . 39

3.4 Set of Simulations . . . 46

4 Results and Discussion Mass Segregation Evolution 48 4.1 Mass Segregation Results . . . 48

4.1.1 NO-SEG . . . 48

4.1.2 SEG-IN . . . 51

4.1.3 SEG-OUT . . . 53

4.2 Summary & Discussion . . . 55

5 Conclusions: Mass segregation 59 5.1 MS Conclusions . . . 59

6 Survival of Embedded Star Clusters 61 6.1 Aim . . . 61

6.2 Previous work . . . 61

6.3 Method . . . 66

6.4 Set of Simulations . . . 68

7 Results and Discussion Survival of Embedded Star Clusters 70 7.1 Survival Results . . . 70

7.1.1 LSF vs fb after 9 Myr, embedded in an uniform BG . . . 70

7.1.2 LSF vs fb after 9 Myr, embedded in a Plummer BG . . . 72

7.1.3 LSF vs fb after 9 Myr, with Qf∼0.5 . . . 74

7.2 Dependency of fb on the time evolution . . . 75

7.2.1 LSF vs fb after 4 Myr, embedded in a uniform BG . . . 75

7.2.2 LSF vs fb after 4 Myr, embedded in a Plummer BG . . . 75

7.2.3 LSF vs fb after 4 Myr, with Qf∼0.5 . . . 78

7.2.4 LSF vs fb after 2 Myr, embedded in an uniform BG . . . 79

7.2.5 LSF vs fb after 2 Myr, embedded in a Plummer BG . . . 79

7.2.6 LSF vs fb after 2 Myr, with Qf∼0.5 . . . 82

7.3 Dependency onΛMST . . . 83

7.4 Evaporation . . . 85

7.4.1 Time vs fb−fbexp, with Qinit= 0.5 and uniform BG . . . 85

7.4.2 Time vs fbexp−fb, with Qinit= 0.2 and uniform BG . . . 85

7.4.3 Time vs fbexp−fb, with Qinit= 0.5 and Plummer BG . . . 87

7.4.4 Time vs fbexp−fb, with Qinit= 0.2 and Plummer BG . . . 88

7.4.5 Time vs fbexp−fb, for Equal-mass simulations . . . 89

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7.5.1 Global Qf as a predictor of fb after 9 Myr . . . 91

7.5.2 Bound stars Qfb as a predictor of fb, after 9 Myr . . . 92

7.5.3 All stars Qfg as a predictor of fb after 9 Myr . . . 93

7.5.4 JoiningΛMST and Qfg as a predictor of fb after 9 Myr . . . 94

7.5.5 JoiningΛMST and Qfg as a predictor of fb after 4 Myr . . . 96

7.6 Discussion . . . 98

8 Conclusion: Survival of Embedded Star Clusters 100

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List of Figures

1.1 OB-association in the LMC called LH 72. Is possible to observe a few high-mass, very young stars embedded in hydrogen gas. Image taken from http://spacetelescope.org/images/potw1147a/ . . . 3 1.2 Optical (left) and Infrared (right) images of the RCW 38 region obtained

with the ESO VLT. The infrared observations can reveal the obscured inner parts of the embedded SC which is not possible with the visual approach. Image taken from https://www.eso.org/public/images/eso0929c/ 4 1.3 An example of a typical open cluster is Pleiades. Image taken from

http://www.rc-astro.com/photo/id1106.html . . . 5 1.4 M80 (NGC 6093) is one of the densest known globular clusters in the Milky

Way Galaxy; it lies in the constellation Scorpius and was discovered by Messier & Niles (1981). Image taken from http://hubblesite.org/image/837/

news_release/1999-26 . . . 7 1.5 Turn o point at dierent ages of SCs. Left panel is the tipical H-R

diagram for an open SCs. Right panel shows the H-R diagram observed for GCs. From Brooks/Cole Thomson Learning. . . 8 1.6 IMF shapes for dierent types of objects. Image taken from Bastian et al.

(2010) . . . 11 1.7 Substructured distribution with stars randomly drawn from an initial mass

function and placed randomly in the spatial distribution. The ten most massive stars are shown by the larger (red) points. Image taken from Parker & Goodwin (2015). . . 16 1.8 Three separate measures of mass segregation for the stellar distribution

shown in Fig. 1.7. In top panel is showed the cumulative distribution (MMF) of the distance from the centre for the ten most massive stars (the

red dashed line) and the cumulative distribution for all stars (the solid black line). In the middle panel is showed theΛMST value as a function

of the NMSTi.e. stars used in the subset. In bottom panel is showed local stellar surface density versus stellar mass (P

LDR). The median stellar surface density for the ten most massive stars is shown by the solid (red) horizontal line and the median surface density for all of the stars is shown by the blue horizontal dashed line. Images taken from Parker & Goodwin (2015). . . 17 1.9 Substructured distribution with stars drawn from an initial mass function

and placed preferentially centred in the spatial distribution. The ten most massive stars are shown by the larger (red) points. Image taken from Parker & Goodwin (2015). . . 19

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1.10 Three separate measures of mass segregation for the stellar distribution shown in Fig. 1.9. In top panel is showed the cumulative distribution (MMF) of the distance from the centre for the ten most massive stars (the

red dashed line) and the cumulative distribution for all stars (the solid black line). In the middle panel is showed theΛMST value as a function

of the NMSTi.e. stars used in the subset. In bottom panel is showed local stellar surface density versus stellar mass (P

LDR). The median stellar surface density for the ten most massive stars is shown by the solid (red) horizontal line and the median surface density for all of the stars is shown by the blue horizontal dashed line. Images taken from Parker & Goodwin (2015). . . 20 1.11 Substructured distribution with stars randomly drawn from an initial mass

function and placed in locations with high local surface densities. The ten most massive stars are shown by the larger (red) points. Image taken from Parker & Goodwin (2015). . . 22 1.12 Three separate measures of mass segregation for the stellar distribution

shown in Fig. 1.11. In top panel is showed the cumulative distribution (MMF) of the distance from the centre for the ten most massive stars (the

red dashed line) and the cumulative distribution for all stars (the solid black line). In the middle panel is showed theΛMST value as a function

of the NMSTi.e. stars used in the subset. In bottom panel is showed local stellar surface density versus stellar mass (P

LDR). The median stellar surface density for the ten most massive stars is shown by the solid (red) horizontal line and the median surface density for all of the stars is shown by the blue horizontal dashed line. Images taken from Parker & Goodwin (2015). . . 23 1.13 Path of low mass star evolution. Image taken from (Padmanabhan, 2001). 25 1.14 Path of low mass star evolution. Image taken from (Padmanabhan, 2001). 26 1.15 Observed frequency of naked SCs and Embedded SCs against age showed

by solid line. Expected frequency assuming a constant SF denoted by dashed line. From Lada & Lada (2003) . . . 28 2.1 Dierent samples reproducing Kroupa IMF (2002). Top-left: total mass

10.000 M. Top-right: total mass 5.000 M. Bottom-left: total mass

1.000 M. Bottom-right: total mass 500 M. From this work. . . 30

2.2 Examples of fractal distribution for dierent fractal dimensions. Top-left fractal with D=1.0. Top-right fractal with D=1.6. Bottom-left fractal with D=2.0. Bottom-right fractal with D=3.0. From this work. . . 32 2.3 The top panel shows the normalised curve for the density inside of the

spheres. The middle panel shows the normalised quantity of the dierent spheres. The bottom panel shows the normalised potential produced. Red solid line is for uniform sphere and green dashed line for Plummer sphere. From this work. . . 35 2.4 Snapshots of the process to nd bound particles. Top panel: Example

of particles distribution. Middle panel: Detected clumps. Bottom panel: Bound particles of each clump. The size of each symbol is a direct repre-sentation of its mass. From this work. . . 37 3.1 Samples of IMF used for the simulations in this work. The dierent colors

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3.2 Examples of the three dierent initial distribution of massive stars as-sumed in this work(shown in 2D). Red lled circles are high mass stars and blue points are low mass stars. The sizes of circles and points are proportional to the mass of the stars. Top-left panel: initial fractal with random placements of the masses (NO-SEG). Top-right panel: initial frac-tal with inside-out segregation (SEG-IN), i.e. the highest mass stars are in the centre of the distribution. Bottom panel: initial fractal with outside-in segregation (SEG-OUT), i.e. the highest mass are further away from the centre. From this work. . . 44 3.3 Example of the evolution of the virial ratio for fractal distributions. The

green solid line shows the oscilations for one fractal starting in virial equi-librium (Q= 0.5) and the blue dashed line shows the changes for the same

fractal now starting in a cool state (Q= 0.2). Horizontal line denotes the

virial equilibrium value of Q= 0.5. From this work. . . 45

3.4 Example of one fractal with initial mass distribution SEG-OUT, Qinit =

0.5and uniform BG. Top panel shows the evolution ofΛMSR with the red

dashed. The evolution ofΛMSR−Rmethod is denoted with the blue solid line. The bottom panel compares the evolution of ther8 (blue solid line)

and the value ofRh (green dashed line). From this work. . . 45

4.1 Results for NO-SEG, i.e. the masses are distributed randomly throughout the whole fractal. In contrary to Fig. 3.4 each datapoint is a mean value calculated from ten dierent random realisations. Top row panels show the results for a uniform background and the lower row panels for the Plummer background. Left panels are the simulations with Qinit= 0.5, i.e. starting in 'pseudo-virial equilibrium' and right panels show the simulations where velocities are reduced to obtain Qinit = 0.2. In each panel the top half shows the evolution of ΛMST (red stars and dashed lines) and Λrest (blue

plus signs and solid lines). In the lower halves of the panels we show the evolution of Rh (green dashed line) andr8 (blue solid line). From this work. 50

4.2 Results for SEG-IN, i.e. where all the massive stars are initially within the central 0.5 pc. In contrary to Fig. 3.4 each datapoint is a mean value calculated from ten dierent random realisations. Top row panels show the results for a uniform background and the lower row panels for the Plummer background. Left panels are the simulations with Qinit = 0.5, i.e. starting in 'pseudo-virial equilibrium' and right panels show the simulations where velocities are reduced to obtain Qinit = 0.2. In each panel the top half shows the evolution of ΛMST (red stars and dashed

lines) andΛrest(blue plus signs and solid lines). In the lower halves of the

panels we show the evolution of Rh (green dashed line) andr8 (blue solid

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4.3 Results for SEG-OUT, i.e. where all the massive stars are initially outside the central 1.0 pc. In contrary to Fig. 3.4 each datapoint is a mean value calculated from ten dierent random realisations. Top row panels show the results for a uniform background and the lower row panels for the Plummer background. Left panels are the simulations with Qinit = 0.5, i.e. starting in 'pseudo-virial equilibrium' and right panels show the simulations where velocities are reduced to obtain Qinit = 0.2. In each panel the top half shows the evolution of ΛMST (red stars and dashed

lines) andΛrest(blue plus signs and solid lines). In the lower halves of the

panels we show the evolution of Rh (green dashed line) andr8 (blue solid

line). From this work. . . 54 4.4 Examples of the nal state for the three dierent initial distribution of

massive stars showed in Fig. 3.2. Left panel: Final state of fractal which start with NO-SEG. Central panel: nal state of fractal which start with SEG-IN. Right panel: nal state of fractal which start with SEG-OUT. From this work. . . 58 6.1 Comparison fb in function of SFE from dierent works using

instanta-neously gas expulsion or an slow one. Figure taken from Baumgardt & Kroupa (2007) . . . 63 6.2 SFE against fb. The solid line represents the results from Baumgardt &

Kroupa (2007) and stars are the results from Smith et al. (2011). From Smith et al. (2011) . . . 64 6.3 Results of LSF against fb. From Smith et al. (2011) . . . 64 6.4 LSF against fb split by dierent ranges of Q. From Farias et al. (2015) . . 65 6.5 Samples of IMF used for the simulations in this work. The dierent colours

are to represent the regions of the masses. From this work. . . 67 6.6 An example of the evolution of the virial-ratio for fractal distributions.

The green solid line shows the oscillations for one fractal starting in virial equilibrium (Q = 0.5) and the blue dashed line shows the changes for

the same fractal now starting in a cool state (Q = 0.2). Horizontal line

denotes the virial equilibrium value of Q= 0.5. The arrows show the four

points of gas expulsion. From this work. . . 68 7.1 Both panels show results for simulations with uniform BG after 9 Myr

evolving without gas. Top and bottom panels with Qinit = 0.5 and Qinit = 0.2 respectively. All the boxes show fb as a function of LSF, divided in six intervals of Qf as Farias et al. (2015). Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 71 7.2 Both panels show results for simulations with Plummer BG after 9 Myr

evolving without gas. Top and bottom panels with Qinit = 0.5 and Qinit = 0.2 respectively. All the boxes show fb as a function of LSF, divided in six intervals of Qf as Farias et al. (2015). Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 73

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7.3 Panels show results for simulations after 9 Myr evolving without gas. Top and bottom panels with uniform BG and Plummer BG respectively. Left panels started with Qinit = 0.5 and right panels with Qinit = 0.2. All the panels show fbas a function of LSF, for Qf∼0.5. Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 74 7.4 Both panels show results for simulations with uniform BG after 4 Myr

evolving without gas. Top and bottom panels with Qinit = 0.5 and Qinit = 0.2 respectively. All the boxes show fb as a function of LSF, divided in six intervals of Qf as Farias et al. (2015). Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 76 7.5 Both panels show results for simulations with Plummer BG after 4 Myr

evolving without gas. Top and bottom with Qinit = 0.5 and Qinit = 0.2 respectively. All the boxes show fb as a function of LSF, divided in six intervals of Qf as Farias et al. (2015). Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 77 7.6 Panels show results for simulations after 4 Myr evolving without gas. Top

and bottom panels with uniform BG and Plummer BG respectively. Left panels started with Qinit = 0.5 and right panels with Qinit = 0.2. All the

panels show fbas a function of LSF, for Qf∼0.5. Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 78 7.7 Both panels show results for simulations with uniform BG after 2 Myr

evolving without gas. Top and bottom panels with Qinit = 0.5 and Qinit = 0.2 respectively. All the boxes show fb as a function of LSF, divided in six intervals of Qf as Farias et al. (2015). Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 80 7.8 Both panels show results for simulations with Plummer BG after 2 Myr

evolving without gas. Top and bottom panels with Qinit = 0.5 and Qinit = 0.2 respectively. All the boxes show fb as a function of LSF, divided in six intervals of Qf as Farias et al. (2015). Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 81 7.9 Panels show results for simulations after 2 Myr evolving without gas. Top

and bottom panels with uniform BG and Plummer BG respectively. Left panels started with Qinit = 0.5 and right panels with Qinit = 0.2. All the

panels show fbas a function of LSF, for Qf∼0.5. Our results are denoted by red circles for SEG-IN and blue crosses for NO-SEG. The green solid line is the prediction of Farias et al. (2015). From this work. . . 82 7.10 Inuence of ΛMST over prediction in top panels for 9 Myr and bottom for

2 Myr, after evolving without gas. Left panels for SEG-IN cases and right panels for NO-SEG cases. Green lled squares mean that the predictions agree and red open squares mean that the predictions disagree. From this work. . . 84

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7.11 Time vs average fb−fbexp, for uniform BG, Qinit = 0.5and Qf ∼0.5. The

dierent symbols denote the ranges of LSF and green solid line is denoting the point zero of fbexp. Top panels show the results for NO-SEG cases, with left panel showing insignicant evaporation and right panel showing signicant evaporation. Bottom panel for SEG-IN case with signicant evaporation. From this work. . . 86 7.12 Time vs average fbexp−fb, for uniform BG, Qinit = 0.2 and Qf ∼ 0.5.

The dierent symbols denote the ranges of LSF and green solid line is denoting the point zero of fbexp. Left panel shows the results for NO-SEG cases and right panel for SEG-IN cases, both with signicant evaporation. From this work. . . 86 7.13 Time vs average fb−fbexp, for Plummer BG, Qinit = 0.5and Qf∼0.5. The

dierent symbols denote the ranges of LSF and green solid line is denoting the point zero of fbexp. Top panels show the results for NO-SEG cases and bottom panels for SEG-IN cases. Left panels are showing insignicant evaporation and right panel are showing signicant evaporation. From this work. . . 87 7.14 Time vs average fb−fbexp, for Plummer BG, Qinit = 0.2 and Qf ∼ 0.5.

The dierent symbols denote the ranges of LSF and green solid line is denoting the point zero of fbexp. Top panels show the results for NO-SEG cases, with left panel showing insignicant evaporation while right panel showing signicant evaporation. Bottom panel for SEG-IN case with signicant evaporation. From this work. . . 88 7.15 Time vs average fbexp−fb, with Qf∼0.5for equal-mass simulations. The

dierent symbols denote the ranges of LSF and green solid line is denoting the point zero of fbexp. Top panels have a uniform BG, left panel with Qinit = 0.5 and right panel with Qinit= 0.2. Middle panels have a Plum-mer BG, left panel with Qinit = 0.5and right panel with Qinit= 0.2. The last four panels show the results with no signicant evaporation values. Bottom panels show the results with signicant evaporation values with Qinit = 0.2, left panel with uniform BG while right panel with Plummer BG. From this work. . . 90 7.16 Comparison between fb as a function of Qf (Q using gas and stars). Top

panels show results for simulations with uniform BG. Bottom panels show results for simulations with Plummer BG. Left panels started with Qinit =

0.5 and right panels with Qinit = 0.2. Red circles for SEG-IN and blue

crosses for NO-SEG. From this work. . . 91 7.17 Comparison between fb as a function of Qfb (Q only using bound stars).

Top panels show results for simulations with uniform BG. Bottom panels show results for simulations with Plummer BG. Left panels started with Qinit = 0.5and right panels with Qinit= 0.2. Red circles for SEG-IN and

blue crosses for NO-SEG. From this work. . . 92 7.18 Comparison between fb as a function of Qfg (Q only using stars). Top

panels show results for simulations with uniform BG. Bottom panels show results for simulations with Plummer BG. Left panels started with Qinit =

0.5 and right panels with Qinit = 0.2. Red circles for SEG-IN and blue

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7.19 Comparison between fb as a function of X. Top panels show results for simulations with uniform BG. Bottom panels show results for simulations with Plummer BG. Left panels started with Qinit = 0.5 and right panels with Qinit = 0.2. Red circles for SEG-IN, blue crosses for NO-SEG and

green solid line for the t. From this work. . . 95 7.20 Comparison between fb as a function of X, for equal-mass simulations.

Red empty circles symbols: uniform BG and Qinit = 0.5. Blue lled circles: uniform BG and Qinit = 0.2. Cyan empty squares: Plummer and

Qinit = 0.5. Magenta lled squares: Plummer and Qinit= 0.2 Green solid

line for the t. From this work. . . 95 7.21 Comparison between fb (after 4 Myr) as a function ofX. Top panels show

results for simulations with uniform BG. Bottom panels show results for simulations with Plummer BG. Left panels started with Qinit = 0.5 and

right panels with Qinit = 0.2. Red circles for SEG-IN, blue crosses for

NO-SEG and green solid line for the t. From this work. . . 97 7.22 Comparison between fb (after 4 Myr) as a function of X, for equal-mass

simulations. Red empty circles symbols: uniform BG and Qinit = 0.5. Blue lled circles: uniform BG and Qinit = 0.2. Cyan empty squares:

Plummer and Qinit = 0.5. Magenta lled squares: Plummer and Qinit =

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List of Tables

1.1 Parameters of the dierent types of associations and clusters. From Galac-tic Dynamics (Princeton Series in Astrophysics) and Portegies Zwart et al. (2010) . . . 8 1.2 Parameters of stars depending of their masses. From T. Padmanabhan

-2001 . . . 27 2.1 Data of measurements for Fig 2.4. First column: Clump number

identi-cation. Second and third column: Position x-axis and y-axis respectively. Fourth column: Number of bound particles. Fifth and sixth column: Vi-sual and bound mass respectively. From this work. . . 36 3.1 Summary of properties of IMF samples showed in Fig. 3.1. From this work. 43 3.2 Summary of initial conditions of simulations used in this work. From this

work. . . 47 4.1 Summary of results. The rst, second and third column are giving

infor-mation of the initial conditions: type of BG potential, mass distribution and virial ratio respectively. The fourth and fth column show the nal values forΛMSR and Λrest respectively. The values are calculated by

tak-ing an average value of all measurements between 4 and 5 Myr. The sixth column gives tseg i.e. the information when (tseg) Λrest stays larger than

2, i.e., a signicant level of MS is detected. The last two columns are the nal values for r8 and Rh respectively. Again we calculate them as

averages from all measurements between 4 and 5 Myr. From this work. . . 57 6.1 Summary of properties of IMF samples showed in Fig. 6.5. From this work. 68 6.2 Summary of initial conditions of simulations used in this work. From this

work. . . 69

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En este trabajo investigamos la evolución de la segregación de masa para cúmulos de estrellas jóvenes inicialmente subestructurados con dos diferentes potenciales de fondo, emulando el gas en estados globales viriales y subviriales. Mediante simulaciones de N cuerpos, seguimos la evolución a lo largo de ∼ 5 Myr. Medimos la segregación de

masa usando el método de minimum spanning tree ΛMSR usando todas las estrellas del

cúmulo. También medimos la segregación de masas con una restricción ΛMSR, usando

sólo las estrellas más masivas dentro de los radios de media masa. Usando diferentes condiciones iniciales, encontramos que diferentes potenciales de fondo pueden conducir en la mayoría de los casos aproximadamente al mismo nivel de segregación de masa, independientemente de las posiciones iniciales de las estrellas masivas. Incluso cuando las diferencias en el resultado nal son pequeñas, la evolución interna es muchos casos es muy diferente, dependiendo fuertemente de las condiciones iniciales, que puede acelerar o frenar la evolución de la segregación.

También estudiamos el proceso de mortalidad. Comenzamos con las mismas condiciones iniciales que en el trabajo anterior, pero ahora eliminando el potencial de fondo, imi-tando la expulsión de gas. Dejamos evolucionar los cúmulos estelares sin gas por 9 Myr después de la expulsión de gas para estudiar su supervivencia. Comparamos con una predicción anterior para simulaciones de masas iguales, encontrando que la inclusión de una función de masa inicial produce una disminución en la fracción de la masa ligada de los remanentes debido a los fuertes encuentros de dos cuerpos y efectos de evaporación.

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We investigate the evolution of mass segregation in initially substructured young embed-ded star clusters with two dierent background potentials mimicking the gas in virial and subvirial global states. By means ofN-body simulation we follow the evolution along∼5

Myr. We measure the mass segregation using the minimum spanning tree methodΛMSR

using all the stars of the clusters. We also measure mass segregation with a restricted

ΛMSR, using only the most massive stars inside of the half-mass radii. Spanning a variety

of dierent initial conditions, we nd that dierent background potentials can lead in most of the cases to approximately the same level of mass segregation, independent of the initial positions of the massive stars. Even when the dierences in the nal result are small, the process of the internal evolution in many cases is very dierent, depending strongly on the initial condition, which can speed up or slow down the evolution of mass segregation.

We also study the infant mortality process. We start with the same initial conditions than described above, but now remove the background potential, mimicking gas expulsion. We let the gas-free star clusters evolve for 9 Myr after the gas expulsion to study their survival. We compare with a previous prediction for equal-mass simulations and nd that the inclusion of an initial mass function produces a decrease in the fraction of the bound mass of the clusters due to strong two-body encounters and evaporation eects.

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General Introduction

1.1 Types of Star Cluster

Star clusters (SCs) have been long recognized as important laboratories for astrophysical research. Studies about this kind of objects have played an important role in developing and understanding of the universe. SCs contain statistically signicant samples of stars spanning a wide range of stellar mass within a relatively small volume of space. Also they could be regarded as the fundamental places of star formation (SF). However only describe a cluster for been the precursor of SF process has bias and some restrictions are mandatory.

Dening what is a cluster can be dicult, and often is merely a matter of personal opinion. The criteria to refer to SC or an association can follow some restrictions, e.g. observed stellar mass volume density, where the values are suciently large to keep the group stable against tidal disruption of the galaxy (ρ >0.1M pc−3; Bok (1934)) and

to pass through interstellar clouds ( ρ>1.0M pc−3; Spitzer (1958)). Adams & Myers

(2001) introduced a criterion where a cluster consists of enough stars to ensure a lifetime of more than 108 years (typical lifetime of open clusters), to eject all its members due

to internal encounters, this lower limit is normally called evaporation time (τev) which

for stellar system in virial equilibrium, can be approximated to τev ∼ 102τrelax, with

the relaxation time of the order of τrelax ∼ 0.1NlnN τcross, with τcross the time which takes

a member to travel through the system and N the number of stars it contains (Binney & Tremaine, 1987). The typical value for open SCs is τcross ∼ 106 years, 0.1NlnN ∼ 1,

obtaining a nal value of N ∼35.

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1.1.1 Associations

This classication consists in stars that look like to form a group with low densities, below the limit to call them SCs as dened in Section 1.1. Most of them are still embedded in their primordial molecular gas and, do not appear to be gravitationally bound systems with densitiesρ <0.1M pc−3.

OB-associations: They can contain thousands of O and B stars covering dozens of parsec. An example of this kind of object is LH 72 showed in Fig. 1.1 located in Large Magellanic Cloud (LMC)

T-associations: They contain less objects, around a hundred of low mass stars, mostly of them are pre-main sequence type called T-Tauri.

A summary of important parameters of this kind of objects are showed in Tab. 1.1, column 1 (OB-associations) and column 2 (T-associations).

Figure 1.1: OB-association in the LMC called LH 72. Is possible to observe a few high-mass, very young stars embedded in hydrogen gas. Image taken from http://spacetelescope.org/images/potw1147a/

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1.1.2 Embedded Star Clusters

Embedded clusters are clusters that are fully or partially embedded in interstellar gas and dust. The studies of the physical processes of these objects have been limited because the environment that surrounds them which is obscuring the inner parts. In the last decade, the development of infrared astronomy has improved this situation, providing a better study of embedded SCs within molecular clouds (MC). The evolution of these objects is a complex mix of gas dynamics, stellar dynamics, stellar evolution, and radiative transfer which is not completely understood (e.g. Elmegreen (2007), Price & Bate (2009)), leaving some critical SC properties uncertain.

Lada & Lada (2003) suggest that 70 − 90% of stars form in embedded SCs, which

can be the basic unit of SF. Fig. 1.2 shows how the infrared technology can observe deep into the MC (right panel) in comparison with the visual capabilities (left panel). These kind of clusters are the youngest known stellar systems and can also be considered proto-clusters, with an embedded phase which appears to not last long, between 2 and 3 Myr. Ages older than 5 Myr rarely appear to be associated with a MC (Leisawitz et al., 1989). After the embedded phase due to dierent processes as supernovae, stellar winds and radiation from OB stars, some of the stars remain bound and evolve into open SCs (∼10%), the rest of stars contribute to the Galactic eld.

The summary of the parameters of embedded SCs is showed in Tab. 1.1, column 3.

Figure 1.2: Optical (left) and Infrared (right) images of the RCW 38 region obtained with the ESO VLT. The infrared observations can reveal the obscured inner parts of the embedded SC which is not possible with the visual approach. Image taken from https://www.eso.org/public/images/eso0929c/

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1.1.3 Open Star Clusters

Open SCs are continuously formed in the disk containing population I stars i.e. only young main sequence stars. They can be characterized as exposed clusters with prac-tically negligible gas. Almost all clusters found in standard open cluster catalogs (e.g., Lynga (1987)) fall into this category. The common inheritance of having formed almost simultaneously from the same parent MC make them useful for observations, which help ll color-magnitude diagrams (CMDs) and the classical theory of star evolution can be tested. Additionally, SCs oer the smallest physical scale over which an accurate ap-proach of the mass function (MF) can be made. The mutual gravitational attraction of its individual members, which support the cluster is determined by Newton's laws of motion and gravity, useful for studies of stellar dynamics. Young SCs have been used as tracers of recent episodes of SF in galaxies and in spiral structures, where the gas is still abundant. One of the most famous SC is shown in Fig. 1.3 called The Pleiads or M45. Some properties of these objects are summarized in Tab. 1.1, column 4.

Figure 1.3: An example of a typical open cluster is Pleiades. Image taken from http://www.rc-astro.com/photo/id1106.html

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1.1.4 Globular Clusters

Globular clusters (GCs) contain typically between104and106 population II stars. Many

of them are located within 10 kpc distance from the Galactic Center, but it is possible to nd them further away. Some of them have distances beyond 50 kpc. Their shapes normally appear nearly spherical, as Fig 1.3 shows. They also are characterised by to be dynamically stable. They contain practically no gas, dust or young stars. The origin of a globular cluster is yet not understood. They were formed billions of years ago (∼1010

yr) and are known as relics of the formation of the galaxy itself. A low number of main sequence stars can be observed compared with open SCs. As they are not actually forming in the Milky Way (with a few exceptions), so a direct empirical study of their formation process is not possible with perhaps some exceptions in extragalactic systems which can be observed. Young massive SCs are sometimes referred as young GCs, with life-times similar to many of the old GCs. The distribution of GCs was used to nd the location of the Galactic Center, the existence of a Galactic halo and setting the overall scale of the Galaxy.

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Figure 1.4: M80 (NGC 6093) is one of the densest known globular clusters in the Milky Way Galaxy; it lies in the constellation Scorpius and was discovered by Messier & Niles (1981). Image taken from http://hubblesite.org/image/837/ news_release/1999-26

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Table 1.1: Parameters of the dierent types of associations and clusters. From Galactic Dynamics (Princeton Series in Astrophysics) and Portegies Zwart et al. (2010)

OB T Embedded Open Globular associations associations Cluster Cluster Cluster Core densityρr <1M pc−3 <1M pc−3 >103M pc−3 >103M pc−3 <103Mpc−3

Size >100 pc <100 pc <5 pc <10 pc 10-100 pc

Number of stars 103105 10-100 102 105 103 105 106

Mass M 103−104M ∼102 M <104M >102M <106M

Lifetime <106yr <106yr <105 yr 108 yr 2×1010 yr

Mostly Close Close Dark Embedded Disk Halo Observed GMC MC MC

1.2 Hertzsprung-Russell diagram

The Hertzsprung-Russell (H-R) diagram, shows the relationship between the luminosi-ties and eective temperatures as the panels in the Fig. 1.5 illustrates. The prominent diagonal is called the main sequence. Depending on the object the main sequence can be dierent.In open SCs, the H-R diagram looks like the left panel. For older objects, the top part of the main sequence become shorter (middle panel) and is possible to observe the turno point where the red giants are located. The main sequence can be even much shorter (right panel) for oldest objects as globular clusters showing the turno point more populated.

Figure 1.5: Turn o point at dierent ages of SCs. Left panel is the tipical H-R diagram for an open SCs. Right panel shows the H-R diagram observed for GCs. From Brooks/Cole Thomson Learning.

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1.3 Overview of Star Formation Process

The process starts with the gravitational collapse of a giant MC (GMC) (or part of) where then the stars are formed. This process only starts if the the equilibrium of the cloud is disturbed e.g. a shock coming of a supernovae or a collision with another cloud, which causes a collapse under its self gravity. AS a consequence of the collapse, the cloud is breaking into many fragments. If the fragments of the GMC have more mass than the Jeans mass (Mj) it is not possible to maintain the hydrostatic equilibrium, producing

essentially a free-fall collapse (Jeans, 1902). The fragmentation continues along the cloud, producing a wide range of masses and even very low mass objects (<0.1M).

As a product of the collapse the gas releases gravitational potential energy as heat. As its temperature and pressure increases, the fragment condenses into a rotating sphere of hot gas producing decrease of the collapse. In this stage of the process it is possible to speak of a proto-star.

The gas still surrounding the proto-star keeps falling onto the core, this is a process dominated by accretion. The accretion of gas generates gravitational energy, part of which further heats the core and part of which is radiated away, providing the luminosity of the proto-star.

When the gas on the core reaches a temperature of ∼2000 K the molecular hydrogen (H2)

starts to dissociate and the hydrostatic equilibrium is no longer possible and a renewed phase of dynamical collapse follows, during which the gravitational energy release is absorbed by the dissociating molecules without a signicant rise in temperature. When H2 is completely dissociated into atomic hydrogen (H), helium (He) is restored and the

temperature rises again. Somewhat later, further dynamical collapse phases follow when rst H and then He are ionized at∼104 K. When ionization of the protostar is complete

it settles back into hydrostatic equilibrium at a much reduced radius

Finally, the accretion slows down and its luminosity is now provided by gravitational contraction. The contraction continues, until the central temperature becomes high enough for nuclear fusion reactions. Once the energy generated by H fusion compensates for the energy loss at the surface, the star stops contracting and reaches the main sequence phase.

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1.4 Virial Ratio

The virial ratio (Q) is the fraction of kinetic energyk and the total potential energy Ω

of a cluster:

Q= k

|Ω| (1.1)

Depending of the value of Q important information of the global processes occurring in a SC can be derived.

• Q= 0.5reects that the SC is in virial equilibrium, a very stable conguration • Q< 0.5 means that Ω is dominating and a collapse of the SC is produced. This

state is known as cool or subvirial state.

• Q>0.5is obtained when kis dominating producing an expansion of the SC.

• Q> 1 is giving information about of a unbound object with stars escaping and

spreading.

Observations show that for SCs in the embedded phase the stars appear to be in a dynam-ically cool state (Peretto et al., 2006, 2007; Proszkow & Adams, 2009) and this property is supported by hydrodynamical simulations of star formation (Klessen & Burkert, 2000; Oner et al., 2009; Maschberger et al., 2010).

1.5 The Initial Mass Function

The initial mass function (IMF) is normally assumed as universal taking studies derived from the Solar Neighborhood (Salpeter, 1955; Miller & Scalo, 1979; Kroupa, 2001) which can not be true. Studies of young massive SCs are decient in low-mass stars and/or top-heavy (Smith & Gallagher, 2001), and dierent behaviors can be observed. Many cases impose minimum and maximum stellar masses. Some of the studies exclude masses lower than ∼1M, which are not important for young SCs but extremely aecting the

oldest SCs(>100)Myr, where the lower-mass stellar population is very important, e.g.,

to reproduce accurate relaxation times. The high masses restriction is less important, there are good relations between the total mass of the cluster and the mass of the most massive star (Vanbeveren, 1982; Weidner & Kroupa, 2004) and they evolve fast enough to disappear after a few Myr.

The MFs are usually described by the following equation:

φ(m) = dN

dm ∝m

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where m is the mass of a star, N the number of stars in a range of mass and α is the slope of the relation obtained.

Salpeter (1955) was the rst to introduce the IMF, providing a very good relation between log(dN/dm) connected withlogmof the stars in SCs, with a value ofα =−2.35. There

is not only one description for the slope of the relation. Phelps & Janes (1993) based their study on several open SCs covering a mass range 1-8 M nding agreement with

the Salpeter value with α = −2.4±0.13. There might be exceptions for the Salpeter

slope (e.g. Sanner et al., 1999; Sanner & Geert, 2001; Prisinzano et al., 2003; Kalirai et al., 2003; Pandey et al., 2005). An improvement of the determination of the subsolar IMF was developed by Moraux et al. (2003) in the Pleiades nding an α=−2.7. Other

measurements in the Magellanic Clouds also support the Salpeter slope (e.g. Cayrel et al., 1988; Sagar & Richtler, 1991; Fischer et al., 1991; Bencivenni et al., 1991; Will et al., 1995) where the SCs dier in many physical properties. Excluding the large uncertainties in some individual measurement, it is possible to talk of an universal IMF (Sagar, 2002), at least in the range 1-20 M. Fig. 1.6 shows dierent objects with their respective MF.

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1.6 Internal Processes of Young Star Clusters

The dynamics of SCs can be split in three big steps, the rst is the embedded phase, which it is followed as a gas free evolution, with mass loss taking the important role. Finally the pure dynamical processes are dening the long-term evolution of the SCs.

1.6.1 Timescales

There are two fundamental timescales of a self-graviting system called dynamical time (τdyn) and the relaxation time (τrel).

The time required for a system to nd the equilibrium can be described withτdyn by:

τdyn= GM 5/2

(−4E)3/2 (1.3)

where E ≡ k+ Ω, the total energy of the system. In virial equilibrium (Q = 0.5)

2k+ Ω = 0, assuming the form of Spitzer (1987):

τdyn= GM

rvir3

−1/2

(1.4)

where rvir = GM2|U|2 the virial radius. The necessary time on which two-body encounters

transfer energy between them establishing thermal equilibrium is represented by τrel (Spitzer, 1987) with the equation:

τrel= hv 2i(3/2)

15.4G2¯ lnΛ

N

8 lnNτcross (1.5) wherehv2ithe root-mean-square speed,m¯ the mean mass,ρthe local density andτcross=

R/hvi, with R the radius of the cluster and hvi the average velocity i.e., the necessary

time for one star travel to the center. The value ofΛis0.4N when all stars have the same mass and are homogeneously distributed with an isotropic velocity distribution (Spitzer, 1987). For systems with dierent masses the eective value ofΛ may be much smaller.

Replacing in the last equation the average values for a cluster in virial equilibrium and assuming Rh =rvir (where Rh is the mass radius) it is possible to obtain the

half-mass two body relaxation time (τrh) represented by the equation:

τrh= N

7 lnΛτdyn (1.6)

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After an SC nd some kind of equilibrium atτrel, the stars can show a Maxwellian velocity distribution. Two-body relaxation allows to the stars to exchange energy between them increasing or decreasing their velocities. Some fraction of the stars (s) can have velocities

larger than vesc escaping from the SC, producing a new Maxwellian distribution, with a new tail of stars which will escape again. In this context if is assumed a constant s

inside of the Rh a dissolution time can be estimated as τdis = τrh/s. For a typical SC

density prole s ≈0.033, resulting in a tdis ≈ 30 trh for isolated SCs (Spitzer (1987)).

The fractionecan no be constant for a SCs orbiting a host galaxy. Gieles & Baumgardt

(2008) show thats≈(Rh/rj)(3/2), where the Jacobi radiusrj =

GM

2(VG/RG)2

(1/3)

(King (1962)), with RG and VG the Galactocentric distance and the circular orbital speed

respectively.

1.6.2 Mass Segregation

Mass segregation (MS) has been detected even in embedded star clusters (Lada et al., 1996; Hillenbrand, 1997; Hillenbrand & Hartmann, 1998; Bonatto & Bica, 2006; Chen et al., 2007; Er et al., 2013), this means that the most massive stars are preferentially concentrated (not necessary in the centre). The process to become a segregated cluster can be dynamical (McMillan et al., 2007; Allison et al., 2009; Yu et al., 2011) and fast for cool and substructured clusters (Allison et al., 2010; Parker et al., 2016) reaching MS in ∼1 Myr. Primordial MS is also possible as result of the star formation process

(Zinnecker, 1982; Murray & Lin, 1996; Elmegreen & Krakowski, 2001; Klessen, 2001; Bonnell et al., 2001; Bonnell & Bate, 2006) where the massive stars tend to form in the centers of the star-forming regions because the higher accretion rates in the central parts or as a result of competitive accretion (Larson, 1982; Murray & Lin, 1996; Bonnell et al., 1997). The last primordial state has been used to explain very high levels of MS in short times when only dynamical processes are not fast enough to explain the observed MS (Bonnell & Davies, 1998; Raboud & Mermilliod, 1998).

The pure theoretical dynamics approach is described in Spitzer (1969) with a segregation time (τseg) for a star with massM. The necessary time for a star to travel to the center

is described by the equation:

τseg≈ m¯

M N

8lnN

R

hvi (1.7)

which e.g. for a cluster of R = 1.5 pc, N = 1000, m¯ = 0.5 M and hvi ∼ 2 km s−1

results in atseg ∼1.7Myr, enough time to segregate even in the embedded phase. This

approximation is only measuring the time which takes for one single star travel from the outer part to the centre, as a free fall, which could not be possible for a populated cluster where many other interactions are happening.

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To quantify the level of MS many methods have been published which in some cases produce contradictory results as they do not measure the same thing. The methods tested by Parker & Goodwin (2015) are summarized in the following section.

1.6.2.1 Radial Mass Functions MMF

This method compares the radial distribution of dierent masses (e.g. Sagar et al., 1988; Gouliermis et al., 2004; Stolte et al., 2006; Sabbi et al., 2008; Chavarría et al., 2010). According to this model if a cluster is showing mass segregation, the most massive stars are preferentially towards the center, and low mass stars located in the outer locations. This method shows some disadvantages. A center denition is required which is accept-able in relaxed, virialized and uniform distributions, but not working in substructured star formation regions.The radial bins are very dependent of the radius-range where it is counting stars and is often arbitrarily. Also there is not a clear interpretation about a quantitative description of MS.

1.6.2.2 Minimum Spanning Tree ΛMST

This method was introduced by Allison et al. (2009) and, relates the spatial distribution of the N most massive stars with respect to many samples of randomly chosen N low massive stars. To measure the spatial distribution the method uses a technique called minimum spanning tree (MST), which it is not necessary and centre denition. The MST in this context is basically the minimum path which connects N stars. To use this denition, it is necessary to nd the length of the N most massive stars and several samples using the sameN for low massive stars to compare as follow:

ΛMST= hlaveragei

lsubset ±

σaverage

lsubset (1.8)

wherehlaveragei is the average of the several measurements ofl from the N low massive stars with the associate error σaverage. The length of the MST of the N most massive stars is denoted bylsubset.

The dierent values of ΛMST can give dierent information. If the low mass stars are

distributed in the same way as the most massive stars, one obtains a value ofΛMST∼1.

Whenhlaverageiis bigger thanlsubsetΛMST>1is obtained, which indicates that the most

massive stars are preferentially concentrated, compared to the low mass stars. A high level of MS is denoted by ΛMST >> 1. It is possible that the low mass stars can be

more concentrated, resulting in a value of ΛMST <1. An associated error is calculated

(±σlaverage

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The ΛMST method is measuring MS more according to the classical sense, where the

most massive stars are preferentially concentrated and not necessarily centered.

1.6.2.3 The Local Density Ratio P LDR

This method was presented by Maschberger & Clarke (2011). A measurement of the local surface density (P

= nπr−21

n) of every star is developed to compare with the average

local surface density of theN most massive stars producing a local surface density ratio. This ratio is giving information of detected MS if the most massive stars are located in regions of higher local surface density. This method is also not assuming the existence of a center. The ratio is expressed as follow:

X LDR=

¯

P subset

¯

P all

(1.9)

1.6.2.4 Comparison of the Methods

In order to test the methods Parker & Goodwin (2015) try the methods for three dif-ferents congurations of masses in a substrutured distribution. Starting with a random distributions as shown in Fig. 1.7, with red lled points denoting the ten most massive stars. The methods detect MS as follows:

i) MMF: The Fig. 1.8, top panel, shows the cumulative distribution from the centre

for all particles as a solid line and, with a dashed line for the ten most massive stars. Until 3 pc both distribution follow almost the same path but splitting after that, due there are not massive stars at radii larger 3.2 pc.

ii) ΛMST: The Fig. 1.8, middle panel, shows the level of MS dependent on the number

of the most massive stars. This method detects MS for the ten most massive stars (1.7±0.4) and, a no depreciable level of MS for the 20 most massive stars .

iii) P

LDR: The Fig. 1.8, bottom panel, shows the stellar surface density for each individual mass. The blue dashed line is the average P

for low mass stars (13.1 stars pc−2) and the red solid line the average for the ten most massive stars (13.2

stars pc−2), nding no MS.

For this conguration for an observer is possible to say that even where the are randomly distributed, some level of concentration is more appreciable for the ten most massive stars. The only method which detect a level of MS is theΛMST.

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Figure 1.7: Substructured distribution with stars randomly drawn from an initial mass function and placed randomly in the spatial distribution. The ten most massive stars are shown by the larger (red) points. Image taken from Parker & Goodwin (2015).

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Figure 1.8: Three separate measures of mass segregation for the stellar distribution shown in Fig. 1.7. In top panel is showed the cumulative distribution (MMF) of the distance from the

centre for the ten most massive stars (the red dashed line) and the cumulative distribution for all stars (the solid black line). In the middle panel is showed theΛMSTvalue as a function of

the NMSTi.e. stars used in the subset. In bottom panel is showed local stellar surface density

versus stellar mass (P

LDR). The median stellar surface density for the ten most massive stars

is shown by the solid (red) horizontal line and the median surface density for all of the stars is shown by the blue horizontal dashed line. Images taken from Parker & Goodwin (2015).

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The methods were tested for a centrally concentrated distribution, locating the ten most massive stars in the inner positions of the fractal, as the Fig 1.9 shows. The distribution looks like the classical picture of MS, where the most massive stars are preferentially concentrated in the centre.

i) MMF: The Fig. 1.10, top panel, shows the cumulative distribution from the centre

for the particles as a solid line and, with a dashed line the ten most massive stars. This method detects MS even when was though for uniform distributions.

ii) ΛMST: The Fig. 1.10, middle panel, shows the level of MS dependent of number of

most massive stars. The method reects a high level of MS for the ten most massive stars (5.3±1.0), but decreasing to very low levels when most of the stars are used.

iii) P

LDR: The Fig. 1.10, bottom panel, shows the stellar surface density according to each individual mass. The blue dashed line is the average P for low mass stars and the red solid line the average for the ten most massive stars, resulting in a P

LDR = 0.58, i.e. no MS. This is because that the most centrally located stars are located in areas with relativity low surface density.

From the distribution shown in Fig. 1.9 is possible to appreciate that the massive stars are preferentially concentrated. MMF and ΛMST detect MS and P

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Figure 1.9: Substructured distribution with stars drawn from an initial mass function and placed preferentially centred in the spatial distribution. The ten most massive stars are shown by the larger (red) points. Image taken from Parker & Goodwin (2015).

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Figure 1.10: Three separate measures of mass segregation for the stellar distribution shown in Fig. 1.9. In top panel is showed the cumulative distribution (MMF) of the distance from the

centre for the ten most massive stars (the red dashed line) and the cumulative distribution for all stars (the solid black line). In the middle panel is showed theΛMSTvalue as a function of

the NMSTi.e. stars used in the subset. In bottom panel is showed local stellar surface density

versus stellar mass (P

LDR). The median stellar surface density for the ten most massive stars

is shown by the solid (red) horizontal line and the median surface density for all of the stars is shown by the blue horizontal dashed line. Images taken from Parker & Goodwin (2015).

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As a third test Parker & Goodwin (2015) located the most massive stars in locations with high local surface densities as Fig 1.11 shows. The classical picture of MS now is observed in locations with high surface densities.

i) MMF: The Fig. 1.12, top panel, shows the cumulative distribution from the centre

for all the particles with a solid line and, with a dashed line for the ten most massive stars. This method detects MS, but as this method lacks statistics the level of MS is not well represented.

ii) ΛMST: The Fig. 1.12, middle panel, shows the level of MS dependent of number

of most massive stars. The method reects a level of MS for the ten most massive stars (2.7±0.5), but decreasing to low levels but still detecting values >1 for the

30 most massive stars. iii) P

LDR: The Fig. 1.12, bottom panel, shows the stellar surface density according each individual mass. The blue dashed line is the average P

for low massive stars and the red solid line the average for the ten most massive stars, resulting now due the high local densities where the most massive stars are located, in a P

LDR= 4.9, i.e. detecting MS. This is not a surprise because this was the scenario to which was created matching exactly with the denition for this method.

Is possible to say that for this scenario the three method agree in nd MS, as is expected for observer watching this kind of conguration.

Parker & Goodwin (2015) concluded that the even when ΛMST and P

LDR can detect MS as the classical picture, the last agree only if the local surface density is high around the most massive stars. In consequence only ΛMST is excluding all the issues previously

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Figure 1.11: Substructured distribution with stars randomly drawn from an initial mass function and placed in locations with high local surface densities. The ten most massive stars are shown by the larger (red) points. Image taken from Parker & Goodwin (2015).

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Figure 1.12: Three separate measures of mass segregation for the stellar distribution shown in Fig. 1.11. In top panel is showed the cumulative distribution (MMF) of the distance from

the centre for the ten most massive stars (the red dashed line) and the cumulative distribution for all stars (the solid black line). In the middle panel is showed theΛMSTvalue as a function of

the NMSTi.e. stars used in the subset. In bottom panel is showed local stellar surface density

versus stellar mass (P

LDR). The median stellar surface density for the ten most massive stars

is shown by the solid (red) horizontal line and the median surface density for all of the stars is shown by the blue horizontal dashed line. Images taken from Parker & Goodwin (2015).

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1.6.3 Overview of Star evolution

Stellar evolution is called to the process which a star changes over the course of time. Depending on the mass of the star, normally in the range of 0.1−60 M, its lifetime

can reach from a few million years for the most massive to trillions of years for the least massive. The time of life is directly related with the nuclear-reaction time which produce a nite time of survival, of the order of tnuc ≈1010(M/M)−2.5 (Padmanabhan, 2001).

The mass of the formed star strictly describes the post-main-sequence evolution, with high mass stars characterised by convective cores and low mass stars by radiative cores.

1.6.3.1 Evolution of High Mass Stars

Starting converting in the core H to He by nuclear reactions, following the path 1-2 in Fig. 1.13, until practically turn to a He-core surrounded by a shell of H. This lead to a heavy core which decreases the pressure. To increase the density and temperature the star contracts producing an increasing of the overall star luminosity to reach the point 3. The H still present in the shells stars to burn because of the increasing temperature of the core after the contraction, expanding very fast the shell passing trough 4 and 5. This process going on decreasing the envelope temperature signicantly moving until 6. A convective zone is produced in the outer layer increasing the luminosity reaching 7. The core which was contracting since 2 nally can reach an enough temperature (

∼ 108 K) to start the He burning and stopping the contraction and again starting to

expand. Contrary to the expansion the outer shells start to contract, this last move the star from 7 to 8. The He burning takes the star from 8 to 10 increasing the temperature and luminosity. As in the previous behaviour but now with He, the decreasing of the available quantity of fuel on the core produce decreasing of a temperature reaching 11. The stars respond contracting to increase the temperature and luminosity until 12 where the high temperature of the core produce a He burning in the outer layers until 12. At this point, the shell starts to contract until 14.

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1.6.3.2 Evolution of Low-Mass Stars

For stars with masses, lower than ∼2.3 M the global evolution is produced in a very

slow way. The path of the evolution is showed in Fig. 1.14, starting burning H and a small expansion (1-2) due to lower temperatures in comparison with massive stars. As in the massive case, the core reacts contracting to reach 3 with an almost exhausted core of H. The core remains contracting to produce in 4 the H-burning in the outer layers until 5. With a very degenerate core, a slow cored contraction and a signicant shell expansion are produced reaching high luminosities between 5-8. Enough temperature in the core produced at 8 where the He ash starts followed by the core He burning. The process between 1-2 is repeated now reaching 10. In this range the star in present in the asymptotic giant branch.

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Table 1.2: Parameters of stars depending of their masses. From T. Padmanabhan - 2001

Mass Core Lifetime nal Comment range Fuel stage

M≤0.08 M - - brown dwarf No H burning.

0.08≤M≤0.5 M H - He He burning never starts.

white dwarf

0.5≤M≤2.2 M He 1-15 CO He ash is produced in a degenerate core, liberating He and

Gyr white dwarf N to the ISM.

2.2≤M≤8 M He 107-109 CO He ash is produced in a non degenerate core, liberating He,

years white dwarf C,N,O to the ISM. The lifetime range is107-109years.

8≤M≤10-12 M C,O <108 Supernova Contributing principally to the ISM with He. Heavy elements

years type II remain captured by the collapsed core..

12≤M≤40 M Heavy ∼106 Neutron Contributing to the ISM with O, Ne, Mg, Si, S, Ca. A neutron

elements years star star of∼1.4Mor black hole is produced.

M>40 M Heavy <106 - Evolution poorly understood.

elements years

1.6.3.3 Final-Stage of Stars

The nal stages of the stars are in direct relation with their mass. For very low mass stars, is possible that the He burning could never be produced because of the low level of conversion of H to He. This produces a decrease in the temperature when the H run out. For stars until ∼8 M can reach further steps from H to He, then to carbon and

oxygen producing a more massive core, contracting and increasing the density. If the core loses their hydrostatic equilibrium will lead to a strong collapse. The time in which the complete process is produced for low mass stars is in the order of Gyr contrary to the case of high massive stars when a star die in the order of Myr. The Tab. 1.2 resume some properties for dierent ranges of masses.

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1.7 Infant mortality

In the previous section it have been described that two dierent phases of young SCs, embedded SCs turn to naked open SCs in a non generic way. The process in which the molecular gas is expelled either by energy injected by proto-stellar jets, energy radiations from massive stars or supernovas explosion can remove the remaining gas (see Lada & Lada (2003)). The process is not possible to follow complete observationally, due the long human-scales, only snapshots are available which are used by simulations to ll the gaps.

Lada & Lada (2003) compared the frequency of SCs for each phase, comparing with the age as the Fig. 1.15 shows. Solid line shows the number of embedded SCs and embedded SCs. In the rst bin all the embedded SCs are found. The dashed line is a prediction of the expected number of open SCs with a constant star formation. The discrepancy between the prediction and the frequency observed conrm that ∼4%of the embedded

SCs can emerge from their natal gas.

Figure 1.15: Observed frequency of naked SCs and Embedded SCs against age showed by solid line. Expected frequency assuming a constant SF denoted by dashed line. From Lada & Lada (2003)

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Codes to use

2.1 IMF Generator

To generate dierent samples of IMF for the simulations, the code use an inverse cumu-lative distribution function. The equations of Kroupa (2002) are analytically integrated and then a Monte Carlo method is used to generate random mass samples, from which the IMF distribution is selected. The equations of Kroupa (2002) are divided by mass region as follow:

N(M)∝

    

   

M−0.3 m0 ≤M/M< m1

M−1.3 m1 ≤M/M< m2

M−2.3 m2 ≤M/M< m3

(2.1)

withm0 = 0.01,m1= 0.08,m2 = 0.5,m3 = 50M.

The Fig. 2.1 shows dierent samples for dierent total masses. The three intervals follow the values of the slopes of Kroupa (2002). For these samples we use more than three thousands masses to ensure enough statistic.

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0.1 0.5 1 5 10 0.1 1 10 100 1000 10000 100000 M [M⊙] d N / d M α

1 = 0.31

α

2 = 1.32

α

3 = 2.26

0.1 0.5 1 5 10

0.1 1 10 100 1000 10000 100000 M [M⊙] d N / d M α 1 = 0.34

α 2 = 1.28

α 3 = 2.23

0.1 0.5 1 5 10

0.1 1 10 100 1000 10000 100000 M [M⊙] d N / d M α 1 = 0.30

α 2 = 1.40

α 3 = 2.09

0.1 0.5 1 5 10

0.1 1 10 100 1000 10000 100000 M [M⊙] d N / d M α 1 = 0.30 α

2 = 1.32 α

3 = 2.22

Figure 2.1: Dierent samples reproducing Kroupa IMF (2002). Top-left: total mass 10.000 M. Top-right: total mass 5.000 M. Bottom-left: total mass 1.000 M. Bottom-right: total

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2.2 Fractal Generator

Advances in observations have shown that star clusters form in a substructured way (Larson, 1995; Elmegreen, 2000; Testi et al., 2000; Williams, 2000; Cartwright & Whit-worth, 2004; Gutermuth et al., 2005; Schmeja & Klessen, 2006; Carpenter & Hodapp, 2008; Schmeja et al., 2008), and this has also been supported by simulations (Klessen & Burkert, 2000, 2001; Bate et al., 2003; Bonnell et al., 2003; Bate, 2009; Oner et al., 2009)

Goodwin & Whitworth (2004) introduced a method to generate initial sub-structured distributions. This method denes a cube of sizeNdiv = 2of which the fractal is created.

Starting in the center of the cube with a rst-generation parent which is divided inNdiv3 sub-cubes. Each sub-cube, called child, can turn into a parent for the next generation with a probability of Ndiv(D−3) where D is the fractal dimension. The probability to become a parent is clearly ruled by the value of the fractal dimension, for lower D less children turn to parents so a more sub-structured distribution is produced. A value of D = 3 correspond to a uniform distribution. The not surviving children are removed

from the box, then a small noise is added to the remaining survivors moving them very close around the actual position, to prevent an articial grid-like structure and nally they become parents for the next step. The mature children or new parents are divided intoNdiv(3) new children each of whichNdiv(D−3), the same probability than before, become a parent. The process is repeated until the number of children reaches a larger number than the number of particles. The velocities of the children are inherited of the parents including a random component that decreases with each new generation. The velocities of the parents follow a Gaussian with a mean of zero and the children inherit the velocity from the parent including a random component that decreases with each new generation Examples of dierent fractals, changing only the fractal dimension are shown in Fig. 2.2, in 2D, for a better appreciation. Top panel shows a fractal with a value of D= 1.0 i.e.

a very clumpy distribution. The value of Dis increasing to the bottom until produce a uniform distribution, which correspond to a value of D= 3.0.

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Figure 2.2: Examples of fractal distribution for dierent fractal dimensions. Top-left fractal with D=1.0. Top-right fractal with D=1.6. Bottom-left fractal with D=2.0. Bottom-right fractal with D=3.0. From this work.

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5 -1 -0.5 0 0.5 1 1.5

y [pc]

x [pc]

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5 -1 -0.5 0 0.5 1 1.5

y [pc]

x [pc]

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5 -1 -0.5 0 0.5 1 1.5

y [pc]

x [pc]

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5 -1 -0.5 0 0.5 1 1.5

y [pc]

Figure

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Referencias

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